Universal Emergence of PageRank

The PageRank algorithm enables to rank the nodes of a network through a specific eigenvector of the Google matrix, using a damping parameter $α\in ]0,1[$. Using extensive numerical simulations of large web networks, with a special accent on British University networks, we determine numerically and analytically the universal features of PageRank vector at its emergence when $α\rightarrow 1$. The whole network can be divided into a core part and a group of invariant subspaces. For $α\rightarrow 1$ the PageRank converges to a universal power law distribution on the invariant subspaces whose size distribution also follows a universal power law. The convergence of PageRank at $α\rightarrow 1$ is controlled by eigenvalues of the core part of the Google matrix which are extremely close to unity leading to large relaxation times as for example in spin glasses.

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